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Question
For what value of λ are the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?
Solution
\[\text{ We have }\]
\[ \vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k} \]
\[\text{ Given that } \vec{a} \text{ and } \vec{b} \text{ are perpendicular }.\]
\[ \Rightarrow \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left( 2 \hat{i} + \lambda \hat{j} + \hat{k} \right) . \left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right) = 0\]
\[ \Rightarrow 2 - 2\lambda + 3 = 0\]
\[ \Rightarrow 5 - 2\lambda = 0\]
\[ \therefore \lambda = \frac{5}{2}\]
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